Positioning Students as Mathematically Competent
Promoting and valuing students’ participation in mathematical discourse—sharing their reasoning; creating, critiquing, and revising arguments; and engaging in collaborations aimed at making sense of and using mathematical ideas—is a way of positioning them as being mathematically competent. In order to ensure that each and every student not only understands and can make use of foundational mathematics concepts and relationships but also comes to experience the joy, wonder, and beauty of mathematics, we must position each and every student as mathematically competent. This requires creating classroom structures—norms and routines—that support students to take risks to engage in discourse and to see themselves as capable and worthy of being heard. In doing so, students’ mathematical identities are connected to their participation in a set of productive practices and processes of doing mathematics. Aguirre, Mayfield-Ingram, and Martin (2013) define mathematical identity as “the dispositions and deeply held beliefs that students develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics in powerful ways across the contexts of their lives” (p. 14).
However, in too many mathematics classrooms mathematical competence is assigned solely on the basis of quickness and correctness, giving the mistaken impression that only some students are “good at math.” This creates an environment where students’ mathematical reasoning goes unexamined and unvalued; consequently, little is known about how they make sense of mathematics, how they use their mathematical understanding in developing solutions, and why their solutions do or do not make sense. Correct answers matter but not as indicators of who is able to do mathematics. Engaging in mathematical discourse is essential for developing mathematical identity and should be recognized as a better indicator of mathematical competence. In what ways must our classrooms and lessons change to promote positive mathematical identities for each and every student?
To get an understanding of positioning students as competent, I invite you to watch Video Clip One from the Bike and Truck Task found in NCTM’s Principles to Actions Toolkit. The video clip is drawn from a high school algebra 1 class but practices modeled in this lesson have strong relevance across all grade bands. To set up the discussion the teacher, Ms. Shackelford, invented a fictional student, Chris, to help her students focus and clarify their thinking about the graphical representation of the position of the truck as a function of time. After watching the video, think about the following questions:
- What norms and routines must have been established and practiced to allow students to engage in the sort of mathematical discourse that positioned each of them as mathematically competent?
- How do the forms of participation move the students forward in their thinking about the mathematics?
- What would happen to the students’ respective mathematical identities if this same task unfolded in a different classroom in which Jacobi is told immediately his reasoning is incorrect and Charles is told he is correct?
The video clip is an illustration of how Ms. Shackelford engaged students in reasoning and sense making through a routine of listening to and critiquing others’ reasoning. Ms. Shackelford positioned Jacobi (yellow shirt) and Charles (maroon shirt) as capable contributors to mathematical discussion. Jacobi’s reasoning did not fit the graphical representation of the truck but he was highly participatory and was able to interact with Charles, whose reasoning did fit the graphical representation of the truck. In the clip we see Jacobi and Charles engaged in public sense-making by sharing their mathematical thinking with their peers and Ms. Shackelford. By publicly making the interaction between Jacobi and Charles worthwhile, Ms. Shackelford positions both students as having mathematical competence through their participation. Their ideas were welcomed and used to build mathematical understanding. When students share and value their mathematical ideas through processes of mathematical discourse, they move away from mathematics competence as producing correct answers quickly and toward mathematics competence as participatory.
Ms. Shackelford conducted this lesson in April of the academic year, and it appeared that the social norm in her classroom had been firmly established and that her students were well aware of, and comfortable with, her expectations that they would explain their thinking, respectfully critique others’ reasoning, and make mathematical connections. The lesson in Ms. Shackelford’s classroom also modeled intellectual authority as being shared between the teacher and students. As students author ideas, decide and justify whether particular ideas are reasonable, and press one another for explanations, they take on forms of intellectual authority that support collaborative mathematics teaching and learning (Langer-Osuna, 2017).
Positioning students as mathematically competent must happen with clarity and consistency to have a long-lasting positive impact on their mathematical identities (Munson, 2018). The questions below are a start for reflecting on how students might be positioned as mathematically competent in your classroom.
- How do I create classroom norms and routines that support students to take risks to engage in mathematical discourse?
- In what ways are students’ mathematical ideas shared and valued?
- How do my teaching practices communicate to each and every student that their ideas matter?
- In what ways is intellectual authority distributed in my classroom?
- How do my teaching practices use students’ ideas to guide them to important mathematical insights and understandings?
I encourage you to use the questions to reflect on your classroom and teaching practices. Please share your successes and challenges on MyNCTM.org.
Robert Q. Berry, III
NCTM President
References
Aguirre, Julia, Karen Mayfield-Ingram, and Danny Martin. The impact of identity in K-8 mathematics: Rethinking equity-based practices. Reston, VA: National Council of Teachers of Mathematics, 2013.
Langer-Osuna, Jennifer M. "Authority, identity, and collaborative mathematics." Journal for Research in Mathematics Education 48.3 (2017): 237-247.
Munson, Jen. "Two Instructional Moves to Promote Student Competence." Teaching Children Mathematics 24.4 (2018): 244-249